What is the field of mathematics that goes deep in foundations

anything pertaining to discrete math, analysis, real analysis...
I mean you can go deep pretty much anywhere. What specifically interests you?

>>8776338
Pajeets are just not meant for deep mathematics.
>inb4 Ramanujan
He's the exception that proves the rule.

>>8777425
The very foundation of real numbers, algorithms that make the basis of mathematics, just like what this book does. Is there a field of mathematics just for that?

>>8777597
Set theory or first-order logic.

Nicolas Bourbaki's first book might interest you.

>>8777482
>Terence Tao
>Pajeet

>talks about mathematics
>exception that proves the rule

Fucking kek. How do you manage with an IQ of 80? Do you have a manual on you that tells you how to breathe? Please respond, writing a thesis on brainlets.

>>8777610
>Nicolas Bourbaki's first book might interest you.
No.
>>8765533

>>8777597
Set theory - jech

File: 2017-03-25 17_33_10-Foundations of Analysis (AMS Chelsea Publishing)_ Edmund Landau_ 9780821826935_ .png (60KB, 393x411px) Image search: [Google]

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>>8777751
Just no.

I would advise "Foundations of Analysis" by Landau.

>>8777632
>Hindustan book agency
OP is a Pajeet, I guarantee it.

>>8777851
I picked the image from amazon. https://www.amazon.com/Analysis-Third-Texts-Readings-Mathematics/dp/9380250649/ref=sr_1_1?s=books&ie=UTF8&qid=1490475898&sr=1-1&keywords=terence+tao+analysis

>>8777751
>>8777610

Set theory and bourbaki are good as practice in semantics, very wordy and philosophical, but are not designed for algorithmic construction of mathematics. If you want to learn about algorithms in depth you need to learn about computer architecture and the actual physical limitations of circuits and how that relates to the theoretical limitations of universality. A good exercise to start is to get comfortable with ASM, and then work your way down into binary, albeit beginning with simple practical circuits and then working your way up towards more complex ones per what you can manage. Graph theory and some baby particle physics are necessary supplements to get an actual foundational view of mathematics.

>>8778087
up

>>8777851
>>8777632
>>8776338
Why the fuck
>Hindustan book agency
publish this and not some repected like Wiley or Pearson?

>>8778827
I don't know, but it is from terence tao

>>8778827
the third edition is a springer book now, so it has a "respected" publisher now

although the Hindustan edition is probably better fucking quality anymore

>>8777805
So cool.
Have you any reference where complex numbers are defined rigorously through the intuitive
[math]i:=\sqrt{-1}[/math]
and then defining operations, and not immediately as ordered reals?

>>8778827
You're a dense motherfucker.

>>8777916
>ASM
https://en.m.wikipedia.org/wiki/Algorithmic_state_machine
?

>>8777805
>Just no.
How so?
Isn't Jech a great reference?

>>8778914
Any analysis text worth its salt (e.g., Rudin) will define the complex numbers as ordered pairs of reals equipped with complex addition and multiplication operations that make C a field.
The ordered reals are simply the reals equipped with a binary relation <=, which is set-theoretically a collection of ordered pairs of reals, that relate the field operations on R to the field operations on C (since relations are generalized functions that need not be left-total or right-unique).
So any "rigorous" definition of i in terms of operations will be isomorphic to the ordered real definition, in that all calculations and theorems of the former will be valid if and only if they are valid in the ordered real definition.

>>8778914
I've not yet seen a text that define's i without using an ordered tuple.

maybe an algebraic completion? but even that's more complicated than what you wrote.

>>8777805
Landau's a great book, but it's not exactly foundations of mathematics.
More like foundations of the number system you use every day.

>>8778941
I already know this but I would be happy to find any reference that completely defines them in an intuitive manner first.
Say, define [math]i=\sqrt{-1}[/math] and consider [math]\mathbb{C}[/math] being [math]\mathbb{R}[/math] along with any possible operation (that we will define rigorously) between elements of [math]\mathbb{R}[/math] and [math]i[/math].
This is what we essentially do when using [math]\mathbb{C}[/math].

File: analysis-ii-3-e-original-imaefsj6asdmzcpz.jpg (17KB, 451x704px) Image search: [Google]

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How do I order this from India?
https://www.flipkart.com/analysis-ii-3-e/p/itmefubshzesa3pq?pid=9789380250656&srno=s_1_1&otracker=search&lid=LSTBOK9789380250656POBSM2&qH=6f444a1dd346fd4c
That's ~ $ 5 but they don't ship abroad.

File: insidelarryellisonstahoepad.jpg (259KB, 958x637px) Image search: [Google]

259KB, 958x637px

>>8778961
OK, I think there's no better alternative to https://www.abebooks.com.
The book is US$ 15.10 complete with shipment to Italy.

>>8778953
This makes the complex numbers a purely algebraic object. It's not clear from this they're truly connected to the plane, and it us thus less powerful. You'll therefore have a hard time finding a book which does that.

how can i get this for free familia?

>>8779036
>You'll therefore have a hard time finding a book which does that.
Is there ANY reference on the internet that does it?
I could try and do this myself; I suspect it isn't very difficult.
Just define set and operations and proceed to verify it's a field but I'd prefer to see something not written by a brainlet.

>>8779039
>how can i get this for free familia?
On libgen you have the Springer edition.

>>8779052
thanks. currently in my first year of math phd and i am sucking fat ass

>>8779049
Thinking about it more, I'm not sure such an approach can be reasonably done, because it's not clear which rules of algebra should be kept and which removed. For instance, under the approach you proposed you'd be tempted to conclude:
1 = sqrt(1) = sqrt((-1)(-1)) = sqrt(-1)sqrt(-1) = i^2 = -1.
You'd need to make a lot of arbitrary rules about which manipulates are valid and which aren't from this naive approach. It's fundamentally much worse than the standard approach, and not worth your time or effort.

>>8776338
I did both his Analysis I book, Baby Rudin and then this http://bass.math.uconn.edu/real.html

There's plenty of foundations of mathematics books or insanely in depth math books like Hardy's Number Theory book

>>8778936

assembly language.

>>8778953
if you define [math]i=\sqrt{-1}[/math] then you run into the problem eventually that [math]-i[/math] is also a solution to it:

Suppose [math]\sqrt{-1}=x+iy[/math].
Then:
[eqn]-1=(x+iy)^2=x^2-y^2+2ixy[/eqn]
Comparing real and imaginary parts, we get [math]xy=0[/math] and [math]x^2-y^2[/math]. Since no real number squared gives [math]-1[/math], we get that [math]x=0, y^2=1\Rightarrow y=\pm 1[/math].
Hence [math]\sqrt{-1}=\pm i[/math].

This is why [math]i^2=-1[/math] is the proper definition given.

Also in some textbooks the complex numbers are defined as [math]\mathbb{C}:=\mathbb{R} +i\mathbb{R}[/math] with [math]i^2=-1[/math] (or [math]\mathbb{R}\times\mathbb{R}, [/math] or [math] \mathbb{R}\oplus \mathbb{R}[/math]), and in others, it is defined as [math]\mathbb{R}^2[/math] with a product defined as [eqn](a,b)\cdot (c,d)= (ac-bd, ad+bc)[/eqn]

>>8779608
>>8779926
Thanks.